Topic
K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: It is shown how to find in time O ( k n ) an optimal colouring, amaximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k if a representation of G is given.
3 citations
TL;DR: In this paper, a study is made of non-regular planar 3-connected graphs with constant weight, where the weights of the vertices of the graphs have constant number of vertices.
Abstract: A study is made of the non-regular planar 3-connected graphs with constant weight.
3 citations
01 Jan 2012
TL;DR: The relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function has been studied in detail by Baldoni and Vergne using techniques of residues.
Abstract: We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide combina- torial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula Q n 2 k=0 Cat(k) for the typeAn case, whereCat(k) is thek th Catalan number, we introduce typeC n+1 and Dn+1 Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes. R´
3 citations
20 May 2015
TL;DR: In this paper, the authors considered the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time, and the newly arrived vertex forms a singleton clique, and can merge any existing cliques in its partitioning into larger cliques, but splitting cliques is not allowed.
Abstract: A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quality of a clique clustering is measured by the total number of edges in its cliques. We consider the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time. At each step, the newly arrived vertex forms a singleton clique, and the algorithm can merge any existing cliques in its partitioning into larger cliques, but splitting cliques is not allowed. We give an online strategy with competitive ratio $$15.645$$ and we prove a lower bound of $$6$$ on the competitive ratio, improving the previous respective bounds of $$31$$ and $$2$$.
3 citations
TL;DR: The results seem to suggest that the topology still determines the K -behavior in these cases, and that there are no K -convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle.
3 citations